↑

Properties

Label 2-483-21.20-c1-0-16
Degree $2$
Conductor $483$
Sign $-0.570 - 0.821i$
Analytic cond. $3.85677$
Root an. cond. $1.96386$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.714i·2-s + (−0.852 + 1.50i)3-s + 1.48·4-s − 0.589·5-s + (−1.07 − 0.609i)6-s + (2.38 + 1.14i)7-s + 2.49i·8-s + (−1.54 − 2.57i)9-s − 0.420i·10-s + 3.80i·11-s + (−1.27 + 2.24i)12-s − 2.33i·13-s + (−0.820 + 1.70i)14-s + (0.502 − 0.888i)15-s + 1.19·16-s + 1.72·17-s + ⋯
L(s)  = 1  + 0.505i·2-s + (−0.492 + 0.870i)3-s + 0.744·4-s − 0.263·5-s + (−0.439 − 0.248i)6-s + (0.900 + 0.433i)7-s + 0.881i·8-s + (−0.514 − 0.857i)9-s − 0.133i·10-s + 1.14i·11-s + (−0.366 + 0.648i)12-s − 0.646i·13-s + (−0.219 + 0.455i)14-s + (0.129 − 0.229i)15-s + 0.299·16-s + 0.419·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.570 - 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.570 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.570 - 0.821i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.570 - 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.674397 + 1.28949i\)
\(L(\frac12)\) \(\approx\) \(0.674397 + 1.28949i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.852 - 1.50i)T \)
7 \( 1 + (-2.38 - 1.14i)T \)
23 \( 1 + iT \)
good2 \( 1 - 0.714iT - 2T^{2} \)
5 \( 1 + 0.589T + 5T^{2} \)
11 \( 1 - 3.80iT - 11T^{2} \)
13 \( 1 + 2.33iT - 13T^{2} \)
17 \( 1 - 1.72T + 17T^{2} \)
19 \( 1 - 5.00iT - 19T^{2} \)
29 \( 1 + 2.26iT - 29T^{2} \)
31 \( 1 - 0.103iT - 31T^{2} \)
37 \( 1 + 3.37T + 37T^{2} \)
41 \( 1 - 2.26T + 41T^{2} \)
43 \( 1 + 3.75T + 43T^{2} \)
47 \( 1 + 11.4T + 47T^{2} \)
53 \( 1 + 6.67iT - 53T^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 + 8.82iT - 61T^{2} \)
67 \( 1 - 13.1T + 67T^{2} \)
71 \( 1 - 7.24iT - 71T^{2} \)
73 \( 1 + 8.17iT - 73T^{2} \)
79 \( 1 + 1.35T + 79T^{2} \)
83 \( 1 - 17.1T + 83T^{2} \)
89 \( 1 - 2.13T + 89T^{2} \)
97 \( 1 - 8.67iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.37788956677857450883277072796, −10.37555594606345256077275130000, −9.740465333926721587541315787800, −8.328189067848071633857879515300, −7.76835972365762114349172399140, −6.57183680357189152592817926271, −5.57674491705728167867181517534, −4.89271023283298473207669937219, −3.58627459294465629064981200039, −1.99672588179436127892516299978, 0.981227908197627920041841367632, 2.18170609052751991506951735657, 3.55086925216427749808685185398, 5.02582791836220470987242128818, 6.15181394243297603369033189580, 7.03833885143487200308080323236, 7.77853051112898277926164229882, 8.705664146367626471301335691675, 10.15312026385776945970354890850, 11.20889174413567363078005420254

Graph of the $Z$-function along the critical line